Matrices

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A matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. (A matrix is a tensor of rank 2.) The individual items in a matrix are called its elements or entries.

An

m  × n

matrix

A

is commonly written using box brackets (an alternative notation uses large parentheses instead)

$\mathbf {A} ={\begin{bmatrix}{\begin{array}{cccccc}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{array}}\end{bmatrix}}=\left({\begin{matrix}{\begin{array}{cccccc}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{array}}\end{matrix}}\right).$

An example of a matrix with 3 rows and 5 columns is

${\begin{bmatrix}{\begin{array}{rrrrr}-1&0&7&0&7\\34&-3&-67&0&-7\\9&-5&7&1&19\end{array}}\end{bmatrix}}=\left({\begin{matrix}{\begin{array}{rrrrr}-1&0&7&0&7\\34&-3&-67&0&-7\\9&-5&7&1&19\end{array}}\end{matrix}}\right).$

Matrix operations

Unary matrix operations

Transpose

The transpose of an

m  × n

matrix

A

is the

n  ×  m

matrix

AT

formed by interchanging the rows and columns of

A

, i.e.

$(\mathbf {A} ^{\rm {T}})_{ij}:=\mathbf {A} _{ji},\quad 1\leq i\leq m,\ 1\leq j\leq n.$

Binary matrix operations

Scalar multiplication

The left scalar multiplication

c​A

of a scalar

c

and a matrix

A

is defined as

$(c\mathbf {A} )_{ij}:=c\cdot \mathbf {A} _{ij},\quad 1\leq i\leq m,\ 1\leq j\leq n,$

and the right scalar multiplication

A c

of a matrix

A

and a scalar

c

is defined as

$(\mathbf {A} c)_{ij}:=\mathbf {A} _{ij}\cdot c,\quad 1\leq i\leq m,\ 1\leq j\leq n.$

Matrix addition

The sum

A + B

of two

m  × n

matrices

A

and

B

is calculated entrywise, i.e.

$(\mathbf {A} +\mathbf {B} )_{ij}:=\mathbf {A} _{ij}+\mathbf {B} _{ij},\quad 1\leq i\leq m,\ 1\leq j\leq n.$

The commutativity of the elements entails the commutativity of matrix addition, i.e.

$\mathbf {A} _{ij}+\mathbf {B} _{ij}=\mathbf {B} _{ij}+\mathbf {A} _{ij},\quad 1\leq i\leq m,\ 1\leq j\leq n,$

$\implies$

$(\mathbf {A} +\mathbf {B} )_{ij}=(\mathbf {B} +\mathbf {A} )_{ij},\quad 1\leq i\leq m,\ 1\leq j\leq n.$

Matrix multiplication

The matrix multiplication of an

r  × s

matrix

A

and an

s  × t

matrix

B

is defined as

$(\mathbf {A} \mathbf {B} )_{ik}:=\sum _{j=1}^{s}\mathbf {A} _{ij}\mathbf {B} _{jk},\quad 1\leq i\leq r,\,1\leq k\leq t,$

where the product matrix

A B

is an

r  × t

matrix.

Note that matrix multiplication is noncommutative, i.e.

$\mathbf {A} \mathbf {B} \neq \mathbf {B} \mathbf {A} .$

Square matrices

A square matrix is an

n  × n

matrix

${\begin{bmatrix}{\begin{array}{cccccc}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\cdots &a_{nn}\end{array}}\end{bmatrix}},$

where the entries

ai i , 1   ≤   i   ≤   n,

constitute the main diagonal of

A

.

The identity matrix

The

n  × n

identity matrix

In

is a square matrix that has 1’s along the main diagonal and 0’s for all other entries. This matrix is often written simply as

I

, and is special in that it acts like 1 in matrix multiplication.

Examples:

     

I1  =  1 , I2  =  1   0 0   1 , I3  =  1   0   0 0   1   0 0   0   1 , I4  =  1   0   0   0 0   1   0   0 0   0   1   0 0   0   0   1 , I5  =  1   0   0   0   0 0   1   0   0   0 0   0   1   0   0 0   0   0   1   0 0   0   0   0   1 , ...

Trace

Main article page: Trace

The trace of an

n  × n

square matrix is defined as the sum of the elements on the main diagonal (the diagonal from the upper-left to the lower-right), i.e.

tr (A)  :=    n ∑   i  = 1    ai  i  =  a1​1 + a2 2 + ⋯ + an  n .

Determinant

Main article page: Determinant

The determinant of an

n  × n

square matrix is denoted

| A |

or

det A

. The determinant of a

1 × 1

matrix (a scalar) is defined as (here we avoid the notation

| a |

due to confusion with absolute value)

$\det {\begin{bmatrix}a_{}\end{bmatrix}}:=a_{}.$

The determinant of a

2 × 2

matrix is defined as

$\left|{\begin{matrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{matrix}}\right|:=\det {\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}:=a_{11}a_{22}-a_{12}a_{21}.$

The determinant of an

n  × n

square matrix is defined as

$\left|{\begin{matrix}{\begin{array}{cccccc}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\cdots &a_{nn}\end{array}}\end{matrix}}\right|:=\det {\begin{bmatrix}{\begin{array}{cccccc}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{n1}&a_{n2}&\cdots &a_{nn}\end{array}}\end{bmatrix}}:=\sum _{j=1}^{n}(-1)^{i+j}\,a_{ij}\,M_{ij}=\sum _{i=1}^{n}(-1)^{i+j}\,a_{ij}\,M_{ij},$

where the minor

M i j

is defined to be the determinant of the

(n  −  1) × (n  −  1)

matrix obtained from

A

by removing the

i

 th row and the

j

 th column. The expression

( − 1) i + j M i j

is known as the cofactor of

ai j

.

Adjugate

The adjugate of an

n  × n

square matrix

A

is the transpose of the cofactor matrix

C

of

A

, i.e.

adj (A)  :=  CT,

where

Ci j

is the cofactor of

ai j

.

C i j  :=  (−1) i  +  j M i j , 1 ≤ i ≤ n, 1 ≤   j ≤ n,

and where

M i j

is the minor of

ai j

.

Inverse

The inverse of an

n  × n

square matrix

A

is an

n  × n

square matrix

A  − 1

defined implicitly as

A A  − 1  =  A  − 1 A  =  I n

where

$\mathbf {I} _{n}:={\begin{bmatrix}{\begin{array}{cccccc}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{array}}\end{bmatrix}}$

is the

n  × n

identity matrix (the main diagonal entries being 1, all other entries being 0).

A matrix is invertible (regular) if and only if its determinant is nonzero, otherwise the matrix is noninvertible (singular). Laplace’s formula for the inverse matrix is

A  − 1  =  adj (A) det (A)  , det (A) ≠ 0,

where

adj (A)

is the adjugate matrix of

A

.

Eigenvectors and eigenvalues

Main article page: Eigenvectors and eigenvalues

A nonzero scalar

λ

and a nonzero vector

v

satisfying

A v  =  λ v

are called an eigenvalue and an eigenvector of

A

, respectively. The nonzero scalar

λ

is an eigenvalue of an

n  × n

matrix

A

if and only if

A  −  λ I n

is not invertible, which is equivalent to

det (A − λ I n )  =  0.

See also

• Tensors

• Scalars (tensors of rank 0)
• Vectors (tensors of rank 1)
• Matrices (tensors of rank 2)

External links

• Online Matrix Calculator, bluebit.
• Online Matrix Addition / Subtraction, bluebit.
• Online Matrix Multiplication, bluebit.
• Linear Equations Solver, bluebit.